Pauli equation

The Pauli equation is a nonrelativistic quantum wave equation for a spin-½ particle in electromagnetic fields. It extends the Schrödinger equation by coupling the particle’s spin to its magnetic moment and the field.

Pauli spin matrices: Three 2×2 matrices that represent the components of spin for a spin-½ particle. They encode spin in the equation’s two-component wavefunction.
Spin-½: An intrinsic angular-momentum quantum number whose spin projections are plus or minus one-half in units of ℏ. The equation is built specifically for particles with this spin.
Wolfgang Pauli: An Austrian-Swiss theoretical physicist who formulated the exclusion principle and developed the theory of electron spin. He introduced the equation in 1927 to describe electron spin in quantum mechanics.
Pauli paramagnetism: The weak magnetic response of a system of conduction electrons caused by their spin polarization. It applies the equation’s magnetic spin coupling to electron gases in solids.
Klein–Gordon equation: A relativistic wave equation that describes spin-0 particles. It is a relativistic scalar equation, unlike the Pauli equation’s nonrelativistic spinor dynamics.
Minimal coupling: The prescription for introducing electromagnetic potentials into a particle’s momentum and energy operators. It inserts the electromagnetic vector and scalar potentials into the kinetic and potential terms.
Electromagnetic four-potential: A scalar and vector potential whose derivatives determine the electric and magnetic fields. Its scalar and vector parts enter the particle’s energy and momentum operators.
Electron spin: The intrinsic angular momentum of the electron, with spin quantum number one-half. Its introduction created the need to extend wave mechanics beyond scalar wavefunctions.
Spin–orbit interaction: A coupling between a particle’s spin and its orbital motion, often arising from relativistic effects. Pauli-form Hamiltonians describe this interaction in atomic and condensed-matter systems.
Schrödinger equation with magnetic field: The scalar Schrödinger equation with electromagnetic potentials coupled to a charged particle. It includes orbital magnetic effects but lacks the intrinsic spin coupling added by Pauli.

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Relativistic quantum mechanicsCompared with: It captures electron spin without providing a fully relativistic description.